Equatorially balanced C4-face-magic labelings on Klein bottle grid graphs

Abstract

For a graph G = (V, E) embedded in the Klein bottle, let F(G) denote the set of faces of G. Then, G is called a C k -face-magic Klein bottle graph if there exists a bijection f :We consider the m × n grid graph, denoted by Km,n, embedded in the Klein bottle in the natural way. We show that for m, n ≥ 2, Km,n admits a C 4 -face-magic Klein bottle labeling if and only if n is even. We say that a C 4 -face-magic Klein bottle labeling {x i,j : (i, j) ∈ V (Km,n)} on Km,n is equatorially balanced if x i,j + x i,n+1−j = 1 2 S for all (i, j) ∈ V (Km,n). We show that when m is odd, a C 4 -face-magic Klein bottle labeling on Km,n must be equatorially balanced. Also when m is odd, we show that (up to symmetries on the Klein bottle) the number of C 4 -face-magic Klein bottle labelings on the m × 4 Klein bottle grid graph is 2 m (m − 1)! τ (m), where τ (m) is the number of positive divisors of m.Furthermore, let m ≥ 3 be an odd integer and n ≥ 6 be an even integer. Then, the minimum number of distinct C 4 -face-magic Klein bottle labelings X on Km,n (up to symmetries on a Klein bottle) is either (5 • 2 m )(m − 1)! if n ≡ 0 (mod 4), or (6 • 2 m )(m − 1)! if n ≡ 2 (mod 4).